problem solving fraction examples

Word Problems with Fractions

Today we are going to look at some examples of word problems with fractions.

Although they may seem more difficult, in reality, word problems involving fractions are just as easy as those involving whole numbers. The only thing we have to do is:

• Read the problem carefully.
• Think about what it is asking us to do.
• Think about the information we need.
• Simplify, if necessary.
• Think about whether our solution makes sense (in order to check it).

As you can see, the only difference in fraction word problems is step 5 (simplify) .

There are some word problems which, depending on the information provided, we should express as a fraction.  For example:

In my fruit basket, there are 13 pieces of fruit, 5 of which are apples. 

How can we express the number of apples as a fraction?

5 – The number of apples (5) corresponds to the numerator (the number which expresses the number of parts that we wish to represent).

13 – The total number of fruits (13) corresponds to the denominator (the number which expresses the number of total possible parts).

The solution to this problem is an irreducible fraction (a fraction which cannot be simplified). Therefore, there is nothing left to do.

Word problems with fractions: involving two fractions

In these problems, we should remember how to carry out operations with fractions.

Carefully read the following problem and the steps we have taken to solve it:

What fraction of the payment has Maria spent?

We find the common denominator:

We calculate:

Word problems with fractions: involving a fraction and a whole number

Finally, we are going to look at an example of a word problem with a fraction and a whole number. Now we will have to convert all the information into a fraction with the same denominator (as we did in the example above) in order to calculate

  We convert 1 into a fraction with the same denominator:

What do you think of this post? Do you see how easy it is to solve word problems with fractions?

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Learn More:

• Understand What a Fraction Is and When It Is Used
• Fraction Word Problems: Addition, Subtraction, and Mixed Numbers
• Learn and Practice How to Subtract or Add Fractions
• Learn How to Subtract Fractions
• Review and Practice the Two Methods of Dividing Fractions
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38 Comments

Thanks for your help

it simplifies the teaching and learning process

Thanks for the explanation… really grateful 🙏

Thank you for such good explanations, it helped me a lot

It is really good it helped me improve my math a lot.

same it helps me in my math too

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Good exercises

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Hi can you not show the answer till the bottom of the page or your giving away the answer so if you solved number one problem the number one aware to the question will be there at the bottom of the page because it is way to easy if it is right there

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Hi Letlhogonolo,

Thank you very much for your comment. If you want to learn more content like this and practice elementary school math, just sign up at Smartick . You have a free trial period with no strings attached. If you have any additional questions or doubts you can write to my colleagues of the pedagogical team at [email protected] .

Best regards!

I like it… but you can level up please 🙄

Roll two dices, the first dice is the numerator, the second is the denominator, this is the first fraction. Roll both dices again and repeat the process to generate the second fraction. Write a division story problem that incorporates these two fractions.

Seems easy of the examples but when I have fraction word promblems in front of me then its still hard for me to figure it out.The examples on this site still is helpful.I will use the site that you give on here to get further practice.Thank you for the examples on here

Interesting and very helpful. I’m going to continue using this site and tell others about it too.

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I think it was really good how you are helping fellow students! But I think you can improve if there were more problems for solving! Thanks

Cool, it helps a lot.

it is helpfull

Solving Word Problems by Adding and Subtracting Fractions and Mixed Numbers

Learn how to solve fraction word problems with examples and interactive exercises.

Example 1: Rachel rode her bike for one-fifth of a mile on Monday and two-fifths of a mile on Tuesday. How many miles did she ride altogether?

Analysis: To solve this problem, we will add two fractions with like denominators.

Solution: 

Answer: Rachel rode her bike for three-fifths of a mile altogether.

Analysis: To solve this problem, we will subtract two fractions with unlike denominators.

Answer: Stefanie swam one-third of a lap farther in the morning.

Analysis: To solve this problem, we will add three fractions with unlike denominators. Note that the first is an improper fraction.

Answer: It took Nick three and one-fourth hours to complete his homework altogether.

Analysis: To solve this problem, we will add two mixed numbers, with the fractional parts having like denominators.

Answer: Diego and his friends ate six pizzas in all.

Analysis: To solve this problem, we will subtract two mixed numbers, with the fractional parts having like denominators.

Answer: The Cocozzelli family took one-half more days to drive home.

Analysis: To solve this problem, we will add two mixed numbers, with the fractional parts having unlike denominators.

Answer: The warehouse has 21 and one-half meters of tape in all.

Analysis: To solve this problem, we will subtract two mixed numbers, with the fractional parts having unlike denominators.

Answer: The electrician needs to cut 13 sixteenths cm of wire.

Analysis: To solve this problem, we will subtract a mixed number from a whole number.

Answer: The carpenter needs to cut four and seven-twelfths feet of wood.

Summary: In this lesson we learned how to solve word problems involving addition and subtraction of fractions and mixed numbers. We used the following skills to solve these problems: 

• Add fractions with like denominators.
• Subtract fractions with like denominators.
• Find the LCD.
• Add fractions with unlike denominators.
• Subtract fractions with unlike denominators.
• Add mixed numbers with like denominators.
• Subtract mixed numbers with like denominators.
• Add mixed numbers with unlike denominators.
• Subtract mixed numbers with unlike denominators.

Directions: Subtract the mixed numbers in each exercise below.  Be sure to simplify your result, if necessary.  Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

Note: To write the fraction three-fourths, enter 3/4 into the form. To write the mixed number four and two-thirds, enter 4, a space, and then 2/3 into the form.

Free Mathematics Tutorials

Fractions questions and problems with solutions.

Questions and problems with solutions on fractions are presented. Detailed solutions to the examples are also included. In order to master the concepts and skills of fractions, you need a thorough understanding (NOT memorizing) of the rules and properties and lot of practice and patience. I hope the examples, questions, problems in the links below will help you.

• Fractions and Mixed Numbers , define fractions and mixed numbers, and introduce important vocabulary.
• Evaluate Fractions of Quantities .
• Fractions Rules including questions with Solutions
• Properties of Fractions
• Complex Fractions with Variables
• Equivalent Fractions examples and questions with solutions.
• Reduce Fractions examples and questions with solutions.
• Reduce Fractions Calculator
• Simplify Fractions , examples and questions including solutions.
• Factor Fractions , examples with questions including solutions.
• Adding Fractions. Add fractions with same denominator or different denominator. Several examples with detailed solutions and exercises.
• Multiply Fractions. Multiply a fraction by another fraction or a number by a fraction. Examples with solutions and exercises.
• Divide Fractions. Divide a fraction by a fraction, a fraction by a number of a number by a fraction. Several examples with solutions and exercises with answers.

Fractions Per Grade

• Fractions and Mixed Numbers , questions and problems with solutions for grade 7
• Fractions and Mixed Numbers , questions and problems with solutions for grade 6
• Fractions , questions for grade 5 and their solutions
• Fractions , questions for grade 4 their answers.

Fractions Calculators

• Fractions Calculator including reducing, adding and multiplying fractions.
• Add Mixed Numbers Calculator

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Fraction Worksheets

Conversion :: Addition :: Subtraction :: Multiplication :: Division

Conversions

Fractions - addition, fractions - subtraction, fractions - multiplication, fractions - division.

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How to Solve Fraction Questions in Math

Last Updated: May 29, 2024 Fact Checked

This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Sophia Latorre . Mario Banuelos is an Associate Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,205,325 times.

Fraction questions can look tricky at first, but they become easier with practice and know-how. Start by learning the terminology and fundamentals, then pratice adding, subtracting, multiplying, and dividing fractions. [1] X Research source Once you understand what fractions are and how to manipulate them, you'll be breezing through fraction problems in no time.

How to Solve Fractions

• If two fractions have the same denominator, add or subtract the numerators from each other.
• If the fractions don’t have the same denominator, change them to a common multiple. For example, 4/5 and 3/2 can become 8/10 and 15/10.
• Multiply fractions by multiplying the numerators, then the denominators. Divide fractions by inverting one and then multiplying the new fractions’ numerators and denominators.

Doing Calculations with Fractions

• For instance, to solve 5/9 + 1/9, just add 5 + 1, which equals 6. The answer, then, is 6/9 which can be reduced to 2/3.

• For instance, to solve 6/8 - 2/8, all you do is take away 2 from 6. The answer is 4/8, which can be reduced to 1/2.

• For example, if you need to add 1/2 and 2/3, start by determining a common multiple. In this case, the common multiple is 6 since both 2 and 3 can be converted to 6. To turn 1/2 into a fraction with a denominator of 6, multiply both the numerator and denominator by 3: 1 x 3 = 3 and 2 x 3 = 6, so the new fraction is 3/6. To turn 2/3 into a fraction with a denominator of 6, multiply both the numerator and denominator by 2: 2 x 2 = 4 and 3 x 2 = 6, so the new fraction is 4/6. Now, you can add the numerators: 3/6 + 4/6 = 7/6. Since this is an improper fraction, you can convert it to the mixed number 1 1/6.
• On the other hand, say you're working on the problem 7/10 - 1/5. The common multiple in this case is 10, since 1/5 can be converted into a fraction with a denominator of 10 by multiplying it by 2: 1 x 2 = 2 and 5 x 2 = 10, so the new fraction is 2/10. You don't need to convert the other fraction at all. Just subtract 2 from 7, which is 5. The answer is 5/10, which can also be reduced to 1/2.

• For instance, to multiply 2/3 and 7/8, find the new numerator by multiplying 2 by 7, which is 14. Then, multiply 3 by 8, which is 24. Therefore, the answer is 14/24, which can be reduced to 7/12 by dividing both the numerator and denominator by 2.

• For example, to solve 1/2 ÷ 1/6, flip 1/6 upside down so it becomes 6/1. Then just multiply 1 x 6 to find the numerator (which is 6) and 2 x 1 to find the denominator (which is 2). So, the answer is 6/2 which is equal to 3.

Joseph Meyer

Think about fractions as portions of a whole. Imagine dividing objects like pizzas or cakes into equal parts. Visualizing fractions this way improves comprehension, compared to relying solely on memorization. This approach can be helpful when adding, subtracting, and comparing fractions.

Practicing the Basics

• For instance, in 3/5, 3 is the numerator so there are 3 parts and 5 is the denominator so there are 5 total parts. In 7/8, 7 is the numerator and 8 is the denominator.

• If you need to turn 7 into a fraction, for instance, write it as 7/1.

• For example, if you have the fraction 15/45, the greatest common factor is 15, since both 15 and 45 can be divided by 15. Divide 15 by 15, which is 1, so that's your new numerator. Divide 45 by 15, which is 3, so that's your new denominator. This means that 15/45 can be reduced to 1/3.

• Say you have the mixed number 1 2/3. Stary by multiplying 3 by 1, which is 3. Add 3 to 2, the existing numerator. The new numerator is 5, so the mixed fraction is 5/3.

Tip: Typically, you'll need to convert mixed numbers to improper fractions if you're multiplying or dividing them.

• Say that you have the improper fraction 17/4. Set up the problem as 17 ÷ 4. The number 4 goes into 17 a total of 4 times, so the whole number is 4. Then, multiply 4 by 4, which is equal to 16. Subtract 16 from 17, which is equal to 1, so that's the remainder. This means that 17/4 is the same as 4 1/4.

Fraction Calculator, Practice Problems, and Answers

Community Q&A

• Check with your teacher to find out if you need to convert improper fractions into mixed numbers and/or reduce fractions to their lowest terms to get full marks. Thanks Helpful 3 Not Helpful 1
• Take the time to carefully read through the problem at least twice so you can be sure you know what it's asking you to do. Thanks Helpful 3 Not Helpful 2
• To take the reciprocal of a whole number, just put a 1 over it. For example, 5 becomes 1/5. Thanks Helpful 1 Not Helpful 1

You Might Also Like

• ↑ https://www.sparknotes.com/math/prealgebra/fractions/terms/
• ↑ https://www.bbc.co.uk/bitesize/articles/z9n4k7h
• ↑ https://www.mathsisfun.com/fractions_multiplication.html
• ↑ https://www.mathsisfun.com/fractions_division.html
• ↑ https://medium.com/i-math/the-no-nonsense-straightforward-da76a4849ec
• ↑ https://www.youtube.com/watch?v=PcEwj5_v75g
• ↑ https://sciencing.com/solve-math-problems-fractions-7964895.html

About This Article

To solve a fraction multiplication question in math, line up the 2 fractions next to each other. Multiply the top of the left fraction by the top of the right fraction and write that answer on top, then multiply the bottom of each fraction and write that answer on the bottom. Simplify the new fraction as much as possible. To divide fractions, flip one of the fractions upside-down and multiply them the same way. If you need to add or subtract fractions, keep reading! Did this summary help you? Yes No

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High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

In order to access this I need to be confident with:

Here you will learn about fractions, including different types of fractions, how to compare fractions, and how to operate with fractions.

Students will first learn about fractions as part of numbers and operations – fractions in 4th and 5th grade. They will continue to expand upon this knowledge in the number system in 6th grade.

What are fractions?

Fractions are ways to show equal parts of a whole.

The denominator of a fraction (bottom number) shows how many equal parts the whole has been divided into.

The numerator of a fraction (top number) shows how many of the equal parts there are.

A proper fraction is a fraction where the numerator is smaller than the denominator.

For example,

2 equal parts 4 equal parts 12 equal parts

One-half is shaded three-quarters is shaded seven-twelfths is shaded

Step-by-step guide : Numerator and Denominator

Equivalent fractions are fractions that have the same value . You can use models or multiplication to find equivalent fractions.

What are two fractions equivalent to \, \cfrac{1}{3} \, ?

Step-by-step guide : Equivalent fractions

Two types of equivalent fractions that show wholes and parts are improper fractions and mixed numbers .

An improper fraction is a fraction where the numerator (top number) is larger than the denominator (bottom number).

A mixed number has a whole number and a fractional part . 

Any number greater than 1 can be shown as an improper fraction AND a mixed number.

\cfrac{3}{2} \, is 3 halves, which is one group of \, \cfrac{2}{2} \, and a group of \, \cfrac{1}{2} \, .

1\cfrac{1}{2} \, is 1 and one half, which is 1 whole and a group of \, \cfrac{1}{2} \, .

Step-by-step guide : Improper fraction to mixed number

Step-by-step guide : Mixed number to improper fraction

Comparing fractions is deciding whether one fraction is smaller than, larger than, or equal to another.

To do this, you can use common denominators , common numerators , or compare to benchmark fractions . Use inequality symbols < (less than) and > (greater than) to write the comparison.

Compare \, \cfrac{3}{4} \, and \, \cfrac{3}{8} \, .

Step-by-step guide : Comparing Fractions

[FREE] Fractions Worksheet (Grades 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of fractions. 10+ questions with answers covering a range of 4th, 5th and 6th grade topics to identify areas of strength and support!

Ordering fractions is taking a set of fractions and listing them in ascending order (least to greatest) or descending order (greatest to least).

To do this, you use equivalent fractions to create common denominators and then compare the numerators.

Order \, \cfrac{2}{3}, \, \cfrac{3}{5} \, , and \, \cfrac{5}{6} \, from least to greatest.

All the denominators (3, 5 , and 6) have a multiple of 30, so you can use 30 as the common denominator.

\cfrac{2 \, \times \, 10}{3 \, \times \, 10}=\cfrac{20}{30} \quad \quad \cfrac{3 \, \times \, 6}{5 \, \times \, 6}=\cfrac{18}{30} \quad \quad \cfrac{5 \, \times \, 5}{6 \, \times \, 5}=\cfrac{25}{30}

From least to greatest, the fractions are:

\cfrac{18}{30}, \, \cfrac{20}{30}, \, \cfrac{25}{30} \,\, or \,\, \cfrac{3}{5}, \, \cfrac{2}{3}, \, \cfrac{5}{6} \, .

Step-by-step guide : Ordering fractions

Adding and subtracting fractions is when you operate with two or more fractions to find the difference or the total.

To do this, fractions need a common denominator (bottom number) to add or subtract. Then you add or subtract the numerators (top numbers) and keep the denominator the same.

\cfrac{7}{8}-\cfrac{3}{8}=

The equation is taking \, \cfrac{3}{8} \, away from \, \cfrac{7}{8} \, .

Since the denominators are the same, the parts are the same size. You subtract to see how many parts are left:

There are 4 parts. But what size are the parts? They are still eighths, so the denominator stays the same.

\cfrac{7}{8}-\cfrac{3}{8}=\cfrac{4}{8}

You can solve the addition version of this equation, \cfrac{7}{8}+\cfrac{3}{8}=\cfrac{10}{8} \, or 1 \cfrac{2}{8} \,, by adding the numerators instead of subtracting.

If the fractions have unlike denominators, you use equivalent fractions to create fractions with common denominators, then follow the same steps.

See also : Adding fractions

See also : Subtracting fractions

See also : Adding and subtracting fractions

Fractions of numbers are when we multiply a fraction by a whole number. The word “of” means to multiply.

\cfrac{1}{4} \times 12 is the same \, \cfrac{1}{4} \, of 12.

To solve with the equation, make the whole number an improper fraction: \, \cfrac{12}{1} \, .

Then multiply the numerators and denominators together:

\cfrac{1}{4} \times \cfrac{12}{1}=\cfrac{1 \, \times \, 12}{4 \, \times \, 1}=\cfrac{12}{4}=3

Step-by-step guide : Fraction of a number

Multiplying and dividing fractions is solving a multiplication or division equation where one or more of the numbers is a fraction.

\cfrac{1}{3} \times \cfrac{2}{3}

In the model, \, \cfrac{2}{3} \, is yellow, and \, \cfrac{1}{3} \, is blue. The product is where the fractions overlap in green.

The model shows \, \cfrac{2}{3} \, of \, \cfrac{1}{3} \, , so \, \cfrac{1}{3} \times \cfrac{2}{3} = \cfrac{2}{9} \, .

The equation shows the numerators multiplied together and the denominators multiplied together.

Dividing fractions can be solved by keeping the first fraction, flipping the second fraction, and changing to multiplication.

\cfrac{1}{3} \div \cfrac{2}{3}

Keep the dividend (first fraction): \cfrac{1}{3}

Take the reciprocal of the divisor (flip the second fraction): \cfrac{2}{3} → \cfrac{3}{2}

Change to multiplication: \cfrac{1}{3} \times \cfrac{3}{2}

Multiply the fractions: \cfrac{1}{3} \times \cfrac{3}{2} =\cfrac{3}{6}

\cfrac{1}{3} \div \cfrac{2}{3} = \cfrac{3}{6} \, or \, \cfrac{1}{2}

See also : Multiplying fractions

See also : Dividing fractions

See also : Multiplying and dividing fractions

Common Core State Standards

How does this relate to 4th grade math and 5th grade math?

• Grade 4 – Number and Operations – Fractions (4.NF.A.1) Explain why a fraction \, \cfrac{a}{b} \, is equivalent to a fraction \, \cfrac{(n \, \times \, a)}{(n \, \times \, b)} \, by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
• Grade 4 – Number and Operations – Fractions (4.NF.A.2) Compare two fractions with different numerators and different denominators, for example, by creating common denominators or numerators, or by comparing to a benchmark fraction such as \, \cfrac{1}{2} \, . Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols \, > , \, = , \, or \, < , \, and justify the conclusions, for example, by using a visual fraction model.
• Grade 4 – Number and Operations – Fractions (4.NF.B.3) Understand a fraction \, \cfrac{a}{b} \, with a > 1 as a sum of fractions \, \cfrac{1}{b} \, .
• Grade 4 – Number and Operations – Fractions (4.NF.B.4) Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
• Grade 5 – Number and Operations – Fractions (5.NF.A.1) Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
• Grade 5 – Number and Operations – Fractions (5.NF.B.3) Interpret a fraction as division of the numerator by the denominator ( \, \cfrac{a}{b} \, = a \div b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, for example, by using visual fraction models or equations to represent the problem.
• Grade 5 – Number and Operations – Fractions (5.NF.B.4) Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
• Grade 5 – Number and Operations – Fractions (5.NF.B.7) Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
• Grade 6 – The Number System (6.NS.A.1) Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, for example, by using visual fraction models and equations to represent the problem.

How to work with fractions

There are a lot of ways to work with fractions. For more specific step-by-step guides, check out the fraction pages linked in the “What are fractions?” section above or read through the examples below.

Fractions practice questions

Example 1: improper fraction to a mixed number with a model.

Write the improper fraction \, \cfrac{13}{5} \, as a mixed number.

• Model the improper fraction.

Draw 3 wholes and divide them equally into fifths. Then shade in 13 parts.

2 Count the number of wholes and the fraction left over.

There are 2 wholes (or \, \cfrac{10}{5} \, ) shaded in and there is \, \cfrac{3}{5} \, left over.

3 Write the mixed number.

\cfrac{13}{5}=2\cfrac{3}{5}

This can also be solved by dividing the numerator by the denominator:

13 \div 5=2 \, R \, 3 \, or \, 2 \cfrac{3}{5} \, .

Example 2: compare using common denominators

Compare: \, \cfrac{5}{6} \bigcirc \cfrac{11}{12} \, .

See if the fractions have like denominators.

The fractions do not have the same denominators (bottom numbers).

Make equivalent fractions if needed.

Both denominators have 12 as a multiple. To create a common denominator, multiply the numerator and denominator of \, \cfrac{5}{6} \, by 2.

\cfrac{5}{6}=\cfrac{5 \times 2}{6 \times 2}=\cfrac{10}{12} \quad and \quad \cfrac{11}{12}

Write the answer using the original fractions.

\cfrac{10}{12} \, has 10 parts shaded in, and \, \cfrac{11}{12} \, has 11 parts shaded in.

Since the parts are the same size, \, \cfrac{10}{12} \, is smaller.

So, \, \cfrac{5}{6} \, is smaller than \, \cfrac{11}{12} \, .

You write this as \, \cfrac{5}{6} \, < \, \cfrac{11}{12} \, .

You also could have solved this using common numerators and benchmark fractions.

Since \, \cfrac{5}{6} \, is \, \cfrac{1}{6} \, away from 1 , and \, \cfrac{11}{12} \, is \, \cfrac{1}{12} \, away from 1 , and twelfths are smaller than sixths, \, \cfrac{11}{12} \, is closer to 1 and so it is larger.

Example 3: subtracting fractions with unlike denominators

Solve \, \cfrac{5}{8}-\cfrac{1}{2} \, .

Create common denominators (bottom numbers).

Since \, \cfrac{5}{8} \, and \, \cfrac{1}{2} \, do not have like denominators, the parts are NOT the same size.

A common denominator of 8 can be used.

Multiply the numerator and denominator of \, \cfrac{1}{2} \, by 4 to create an equivalent fraction.

\cfrac{5}{8} \quad and \quad \cfrac{1 \, \times \, 4}{2 \, \times \, 4}=\cfrac{4}{8}

Add or subtract the numerators (top numbers).

Now use the equivalent fraction to solve: \, \cfrac{5}{6}-\cfrac{4}{8} \, .

Since the denominators are the same, the parts are the same size. You subtract to see how many parts are left: 5-4 = 1.

Write your answer as a fraction.

There is 1 part. But what size is the part? It is still an eighth, so the denominator stays the same.

\cfrac{5}{8}-\cfrac{4}{8}=\cfrac{1}{8}

Example 4: proper fractions of numbers

Find \, \cfrac{2}{10} \, of 44.

Convert to a multiplication statement.

\cfrac{2}{10} \times 44

Convert the whole number to an improper fraction.

\cfrac{2}{10} \times \cfrac{44}{1}

Multiply the numerators together and the denominators together.

\cfrac{2}{10} \times \cfrac{44}{1}=\cfrac{2 \, \times \, 44}{10 \, \times \, 1}=\cfrac{88}{10} \,\, or \, 8 \cfrac{8}{10}

Example 5: multiplying a mixed number by a fraction with the algorithm

Solve \cfrac{3}{12} \times 2 \cfrac{1}{4} \, .

Convert whole numbers and mixed numbers to improper fractions.

Convert the mixed number to an improper fraction.

2 \cfrac{1}{4}=\cfrac{9}{4}

Multiply the numerators together.

\cfrac{3}{12} \times \cfrac{9}{4}=\cfrac{27}{}

Multiply the denominators together.

\cfrac{3}{12} \times \cfrac{9}{4}=\cfrac{27}{48}

If possible, simplify or convert to a mixed number.

The numerator is less than the denominator, so the answer is a proper fraction. 27 and 48 have a common factor of 3, so the fraction can be simplified.

\cfrac{27 \, \div \, 3}{48 \, \div \, 3}=\cfrac{9}{16}

So, \, \cfrac{3}{12} \times \cfrac{9}{4}=\cfrac{27}{48} \, or \, \cfrac{9}{16} \, .

Example 6: word problem dividing mixed numbers

To make a bracelet, Jenny needs \, \cfrac{2}{5} \, m of string. How many complete bracelets can be made from 3\cfrac{1}{10} \, m of string?

Create an equation to model the problem .

3 \cfrac{1}{10} \div \cfrac{2}{5}= \, ?

Change any mixed numbers to an improper fraction.

3 \cfrac{1}{10} → \cfrac{31}{10}

Take the reciprocal of (or flip) the divisor (second fraction).

\cfrac{2}{5} → \cfrac{5}{2}

Change the division sign to a multiplication sign.

\cfrac{31}{10} \times \cfrac{5}{2}

Multiply the fractions together.

\cfrac{31}{10} \times \cfrac{5}{2}=\cfrac{155}{20}

\cfrac{155}{20} \, has a common factor of 5.

\cfrac{155 \, \div \, 5}{20 \, \div \, 5}=\cfrac{31}{4}=7 \cfrac{3}{4}

7 complete bracelets can be made.

Teaching tips for fractions

• Fraction work in 3rd grade centers around understanding through models; particularly area models and number lines. To build on this in 4th grade, always have physical or digital models available for students to use when necessary.
• When using worksheets to teach fraction concepts, make sure that students have ample room to solve since there are many different ways that students can solve. Students may use different types of models (including number lines and area models) or use different equations to solve. They need room to show all of this thinking on paper. Even once students begin to use the algorithm for a fraction concept, they will still need adequate room to solve.
• Fraction worksheets can be useful when students are developing understanding, but once students have a successful strategy and can flexibly operate, incorporate math games or real world projects that allow students to operate with fractions in a variety of situations.
• Highlight patterns within and between different fraction topics as students are learning. Encourage students to be pattern-seekers themselves. This will help them make sense of and understand why the algorithms work.

Easy mistakes to make

• Not drawing clear models Models can help students solve fraction problems if they are drawn clearly, but can lead to errors and misconceptions if they are drawn incorrectly. They also become more challenging for students to use as the denominator gets larger. Dividing models into many equal parts leaves room for error. To help avoid this, give students access to physical or digital models that are pre-made.
• Assuming that an improper fraction is smaller than a mixed number Both improper fractions and mixed numbers include wholes, they are just shown in different ways. Improper fractions have wholes included as part of the numerator, so it is not always clear how many wholes there are. This makes it easier to assume that a mixed number is greater, but that is not always true. For example, \cfrac{9}{3} \, is larger than 1 \cfrac{1}{3} \, , because \, \cfrac{9}{3}=3.
• Assuming the fraction with the smallest numbers is the smallest fraction The relationship between the numerator and denominator is what gives a fraction its value, not just thinking about the numerator and denominator separately. For example, \cfrac{3}{5} \, may seem smaller than \, \cfrac{55}{100} \, , since 3 and 5 are smaller numbers, but if you use a common denominator to compare them, you see that \, \cfrac{3}{5} \, is the larger fraction: \cfrac{3 \, \times \, 20}{5 \, \times \, 20}=\cfrac{60}{100} \,\, and \,\, \cfrac{60}{100} \, > \, \cfrac{55}{100} \, .

Practice fractions questions

1. Write the following mixed number as an improper fraction: \, 4\cfrac{11}{12}

Multiply the denominator by the whole number.

12 \times 4=48

Add the product to the numerator and keep the same denominator.

48 + 11 = 59

The new numerator is 59 and the denominator is still 12.

4 \cfrac{11}{12}=\cfrac{59}{12}

2.Order the fractions from greatest to least.

All the denominators (2, 3, 5 , and 10) have a multiple of 30, so you can use 30 as the common denominator.

Here are the equivalent fractions with the common denominator of 30:

Now that the denominators are the same, you can order them from least to greatest by the numerator:

Rewrite the fractions in their original form.

3.Solve 3 \cfrac{5}{10}+4 \cfrac{6}{10} \, .

First, add the whole numbers

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 5 + 6 = 11.

There are 11 parts. But what size are the parts? They are still tenths, so the denominator stays the same.

\cfrac{5}{10}+\cfrac{6}{10}=\cfrac{11}{10} \, or \, 1 \cfrac{1}{10}

Add the whole numbers and fraction together.

7+1 \cfrac{1}{10}=8 \cfrac{1}{10}

4.Find \, \cfrac{1}{3} \, of 18.

\cfrac{1}{3} \, of 18 is \, \cfrac{1}{3} \times 18.

18 as an improper fraction is \,\cfrac{18}{1} \, .

So, \, \cfrac{1}{3} \times \cfrac{18}{1}=\cfrac{18}{3} \, .

To simplify, use the common factor 3.

\cfrac{18 \, \div \, 3}{3 \, \div \, 3}=\cfrac{6}{1}=6

\cfrac{1}{3} \, of 18 is 6.

5. Solve \, 5 \div \cfrac{1}{4} \, .

5 wholes into groups of \, \cfrac{1}{4} \, is 20 groups.

You can also solve this with the algorithm:

Convert the whole number to a fraction: 5 = \cfrac{5}{1}

Take the reciprocal of (or flip) the second fraction: \, \cfrac{1}{4} \rightarrow \cfrac{4}{1}

Change division to multiplication: \cfrac{5}{1} \times \cfrac{4}{1}

Multiply: \cfrac{5}{1} \times \cfrac{4}{1}=\cfrac{20}{1}=20

6. A recipe for cookies calls for \, \cfrac{3}{4} \, cups of sugar. Roy wants to make 3 \cfrac{1}{2} \, recipes of cookies. How many cups of sugar will he need?

4 \cfrac{1}{4} \, cups

2 \cfrac{5}{8} \, cups

2 \cfrac{3}{4} \, cups

4 \cfrac{4}{6} \, cups

Since each recipe has \, \cfrac{3}{4} \, cups of sugar, multiply to solve: \, \cfrac{3}{4} \times 3 \cfrac{1}{2} \, .

3 \cfrac{1}{2}=\cfrac{7}{2}

Then, multiply the numerators together: \, \cfrac{3}{4} \times \cfrac{7}{2}=\cfrac{21}{} \, .

Then, multiply the denominators together: \, \cfrac{3}{4} \times \cfrac{7}{2}=\cfrac{21}{8} \, .

The numerator is greater than the denominator, so the improper fraction can be converted to a mixed number.

\cfrac{21}{8}=2 \cfrac{5}{8}

Roy needs 2 \cfrac{5}{8} \, cups of sugar.

Fractions FAQs

No, these skills do not require students to convert to lowest terms (also known as the simplest form). That said, it is possible that students will be asked to provide an answer in lowest terms. Refer to your state’s standards for specific guidance on when this is appropriate.

Yes, but just like conversions are required when comparing fractions and mixed numbers, other types of numbers usually also need to be converted for comparisons. For all algorithms, it is important that numbers are in the same form before they are compared.

These are both other names for a mixed number and mean the same thing.

The least common multiple can be used to find the least common denominator when creating equivalent fractions. The greatest common factor can be used to efficiently simplify fractions to their lowest terms. However, since LCM and GCF are not introduced until 6th grade, younger students should not be expected to use these skills.

The next lessons are

• Fractions operations
• Types of fractions
• Simplifying fractions

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Word Problems on Fractions: Types & Solved Examples

Word Problems on Fractions: A fraction is a mathematical expression for a portion of a whole. Each portion acquired when we divide the entire whole into parts is referred to as a fraction. When we divide a pizza into parts, for example, each slice represents a fraction of the whole pizza. Fractions are subjected to a variety of operations, including addition, subtraction, multiplication, and division. Fractions are used in many real-life situations.

This article will outline how to construct and solve fraction word problems. Students will come across fraction word problems with answers, fraction problem solving and dividing fractions word problems. It is advisable to practice all the problems thoroughly before attempting the exam. Keep reading to know more about word problems on fractions,, definition, types, solved examples and many more

Definition of Fractions

A fraction is a number that is used to expresses a part per whole. Each part obtained when we divide the whole into several parts is called the fraction.

Example: When we cut an apple into two-part, then each part represents the fraction \(\left(\frac{1}{2}\right)\) of the apple.

A fraction consists mainly of two parts, one is the numerator, and the other one is the denominator. The upper part or topmost part of the fraction is called the numerator, and the bottom part or below part is called the denominator.

We have mainly three types of fractions: proper fractions, improper fractions, and mixed fractions. They are categorised by the relationship between the numerator and denominator of the fractions.

Word Problems on Fractions

The fraction problem solving consist of a few sentences describing a real-life scenario where a mathematical calculation of fraction formulas are used to solve a problem.

Example: Keerthi took one piece of pizza, which is cut into a total of four pieces. Find the fraction of the pizza taken by Keerthi? The fraction of pizza taken by Keerthi \(=\frac{1}{4}\)

Some of the word problems on fractions that uses fraction formula are listed below:

• Word problems on simplification of fractions
• Word problems on addition and subtraction of fractions
• Word problems on multiplication of fractions
• Word problems on dividing fractions
• Word problems on fractions, percentages, decimals.

Word Problems on Simplifications of Fractions

A fraction in which the numerator and the denominator have no common factor other than “one” is said to be the simplest form of fractions.

Example: Divya took \(8\) apples from the bucket of \(24\) apples. Find the fraction of apples taken by the Divya? The fraction of apples taken by Divya \(=\frac{8}{24}\) and its simplest form is \(\frac{1}{3}\)

Word Problems on Addition of Fractions

To add the like fractions (Fractions with the same denominators), keep the denominator the same and add the numerator values of the given fractions.

To add the unlike fractions (fractions with different denominators), convert the denominators of the given fractions equal to L.C.M of their denominators. Now add the numerator value and take the denominator of the resultant as L.C.M.

Example: Sahana bought \(\frac{1}{4} \mathrm{~kg}\) of apples and \(\frac{1}{2} \mathrm{~kg}\) of oranges from the shop. Total how many fruits she bought? The total fruits bought by Sahana \(=\frac{1}{2}+\frac{1}{4}=\frac{1 \times 2+1}{4}=\frac{3}{4} \mathrm{~kg}\)

Word Problems on Subtraction of Fractions

To subtract the like fractions (Fractions with the same denominators), keep the denominator the same and find the difference of the numerator values of the given fractions.

To subtract the unlike fractions (fractions with different denominators), convert the denominators of the given fractions equal to L.C.M of their denominators. Now find the difference of the numerator value and take the denominator of the resultant as L.C.M.

Example: Keerthi travelled \(\frac{2}{5} \mathrm{~km}\) to school. While returning home, she stopped at her friend’s house at a distance of \(\frac{1}{3} \mathrm{~km}\). Find the remaining distance? The remaining distance needs to be travelled \(=\frac{2}{5}-\frac{1}{3}=\frac{(2 \times 3)-(1 \times 5)}{5 \times 3}=\frac{6-5}{15}=\frac{1}{15} \mathrm{~km}\)

Word Problems on Multiplication of Fractions

To multiply the two or more fractions, find the product of numerators of the given fractions and the product of the denominators of the given fractions separately.

Example: Keerthi had \(Rs.10000\), and she had donated \(\frac{1}{10}\) of the money to the Oldage home. How much amount did she donate? The amount Keerthi donated \(=\frac{1}{10} \times Rs.10000= Rs. 1000\)

Word Problems on Division of Fractions

The division of fractions is nothing but multiplying the first fraction with the reciprocal of the second fraction. The reciprocal of the fraction is a fraction obtained by interchanging the numerator and denominator.

Example: The area of the rectangle is \(\frac{15}{4} \mathrm{~cm}^{2}\), whose length is \(\frac{5}{2} \mathrm{~cm}\). Find the width of the rectangle? We know that area of rectangle \(= \text {length} \times \text {bredath}\) And, breadth \(=\frac{\text { area }}{\text { length }}=\frac{15}{\frac{4}{2}}=\frac{15}{4} \times \frac{2}{5}=\frac{3}{2} \mathrm{~cm}\).

Word Problems on Conversion of Fractions to Percentage

We know that percentages are also fractions with the denominator equals to hundred. To convert the given fraction to a percentage, multiply it with hundred and to convert any percentage value to a fraction, divide with hundred.

Example: Keerthi ate \(\frac{2}{5}\) of the pizza. How much percentage of pizza is eaten by Keerthi? The percentage of pizza ate by Keerthi \(=\frac{2}{5} \times 100 \%=40 \%\).

Word Problems on Conversion of Fractions to Decimals

Decimal numbers are the numbers (quotient) obtained by dividing the fraction’s numerator with the given fraction’s denominator. To convert the given decimal to the fractional value by writing the given number without decimals and making the denominator equal to \(1\) followed by the zeroes and number of zeroes equal to the number of decimal places.

Example: Keerthi got \(\frac{1}{10}\) of the price of a T.V. as a discount. Find the discount in decimal. The part of the discount received by a Keerthi as a discount \(=\frac{1}{10}=0.1\)

Solved Examples – Word Problems on Fractions

Q.1. In February \(2021\) , a school was working only three-fourths of the total number of days in the month and the remaining number of days given as holidays. How many days did the school work in the month of February? Ans: The year \(2021\) is a non-leap year. We know that a non-leap has \(28\) days in February month. So, the total number of days \(=28\). Given, the school was working only three-fourths of the total number of days in the month. The number of days school working in February month \(=\frac{3}{4}\) of \(28\). \(=\frac{3}{4} \times 28=21\) days Hence, the school working for \(21\) days in the month of February for the year \(2021\).

Q.2. Keerthi needs \(1 \frac{1}{2}\) cups of sugar for baking a cake. She decided to make \(6\) cakes for her friends. How many cups of sugar did she need for making the \(6\) cakes? Ans: Given, Keerthi needs \(1 \frac{1}{2}\) cup of sugar to make a cake. The total cups of sugar required to make 6 cakes is calculated by multiplying the sugar needed for one cake with the number of cakes that needs to be prepared by Keerthi and is given by \(1 \frac{1}{2} \times 6\) Convert the above-mixed fraction to an improper fraction by multiplying the denominator with the whole and add to the numerator keeping the same denominator as \(1 \frac{1}{2}=\frac{(\text { whole×denominator })+\text { numerator })}{\text { denominator }}=\frac{(1 \times 2)+1}{2}=\frac{3}{2}\) The total cups of sugar needed for making \(6\) cakes \(=\frac{3}{2} \times 6=9\) Hence, Keerthi needs \(9\) cups of sugar to make \(6\) cakes.

Q.3. An oil container contains \(7 \frac{1}{2}\) litres of oil which are poured into \(2 \frac{1}{2}\) litres bottles. How many bottles are needed to fill \(7 \frac{1}{2}\) litres of oil? Ans: Given, a container holds total oil of \(7 \frac{1}{2}\) litres, and the total amount held by each bottle is \(2 \frac{1}{2}\) litres. Consider the number of bottles required is \(x\). From the given question, the total oil in the container is equal to the product of oil in each bottle and the number of bottles required. \(\Rightarrow 7 \frac{1}{2}=x \times 2 \frac{1}{2}\) \(\Rightarrow \frac{15}{2}=x \times \frac{5}{2}\) \(\Rightarrow 15=5 x\) \(\Rightarrow x=\frac{15}{5}=3\) Therefore, \(3\) bottles are required to fill the total oil in the container.

Q.4. A square garden has the area \(\frac{36}{25} \,\text {sq.ft}\). Find the side of the square garden. Ans: Given the area of the square garden is \(\frac{36}{25} \,\text {sq.ft}\). Let the length of the side of the square garden is \(a\) fts. We know that area of the square \( = {\rm{side}} \times {\rm{side}} = {a^2}\) Thus, \(a^{2}=\frac{36}{25}\) \(\Rightarrow a=\sqrt{\frac{36}{25}}=\frac{\sqrt{36}}{\sqrt{25}}=\frac{6}{5}\) feet. Hence, the length of the side of the square garden is \(\frac{6}{5}\) feet.

Q.5. At a party, total \(280\) ice-creams are prepared. Four-seventh of them is eaten by the children. Find the ice-creams eaten by the children. Ans: Total ice-creams prepared \(=280\) Number of ice-creams eaten by children \(=\frac{4}{7}\) of \(280=\frac{4}{7} \times 280=160\) Hence, children ate \(160\) ice-creams.

In mathematics, a fraction is used to represent a piece of something larger. It depicts the whole’s equal pieces. The numerator and denominator are the two elements of a fraction. The numerator is the number at the top, while the denominator is the number at the bottom. The numerator specifies the number of equal parts taken, whereas the denominator specifies the total number of equal parts in the total.

In this article, we have studied the definitions of fractions,  different types of fractions. We also studied the word problems on fractions and their operations. This article gives the word problems on fractions, addition and subtraction of fractions, multiplication of fractions, division of fractions, the simplest form of fractions, conversion of fractions to percentage, decimals etc., with the help of solved examples.

FAQs on Word Problems on Fractions

Here are some of most commonly asked questions on word problems on fractions.

Q.1: How do you solve word problems with fractions?

Ans: To solve word problems with fractions, first, read and write the given data. Write the mathematical form by given data and perform the operations on fractions according to the data.

Q.2: How do you write a fraction division in word problems? 

Ans: The fraction division can be written as keeping the first fraction as it is and multiplying it with the reciprocal of the second fraction.

Q.3: How do you know when to divide or multiply fractions in a word problem?

Ans: To find the product, we need to multiply and to find any one of the quantities, we need to divide.

Q.4: What is an example of a fraction word problem?

Ans: Keerthi ate 40% of the pizza. How much is part of the pizza eaten by Keerthi.

Q.5: What is a fraction?

Ans: A fraction is a number that is used to express a part per whole.

Learn About Conversion Of Fractions

We hope this detailed article on Word Problem on Fractions helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest.

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Word Problems on Fraction

In word problems on fraction we will solve different types of problems on multiplication of fractional numbers and division of fractional numbers.

1.  4/7 of a number is 84. Find the number. Solution: According to the problem, 4/7 of a number = 84 Number = 84 × 7/4 [Here we need to multiply 84 by the reciprocal of 4/7]

= 21 × 7 = 147 Therefore, the number is 147.

2.  Rachel took \(\frac{1}{2}\) hour to paint a table and \(\frac{1}{3}\) hour to paint a chair. How much time did she take in all?

3. If 3\(\frac{1}{2}\) m of wire is cut from a piece of 10 m long wire, how much of wire is left?

Total length of the wire = 10 m

Fraction of the wire cut out = 3\(\frac{1}{2}\) m = \(\frac{7}{2}\) m

Length of the wire left = 10 m – 3\(\frac{1}{2}\) m

            = [\(\frac{10}{1}\) - \(\frac{7}{2}\)] m,    [L.C.M. of 1, 2 is 2]

            = [\(\frac{20}{2}\) - \(\frac{7}{2}\)] m,    [\(\frac{10}{1}\) × \(\frac{2}{2}\)]

            = [\(\frac{20 - 7}{2}\)] m

            = \(\frac{13}{2}\) m

            = 6\(\frac{1}{2}\) m

4. One half of the students in a school are girls, 3/5 of these girls are studying in lower classes. What fraction of girls are studying in lower classes?

Fraction of girls studying in school = 1/2

Fraction of girls studying in lower classes = 3/5 of 1/2

                                                            = 3/5 × 1/2

                                                            = (3 × 1)/(5 × 2)

                                                            = 3/10

Therefore, 3/10 of girls studying in lower classes.

5.  Maddy reads three-fifth of 75 pages of his lesson. How many more pages he need to complete the lesson? Solution: Maddy reads = 3/5 of 75 = 3/5 × 75

= 45 pages. Maddy has to read = 75 – 45. = 30 pages. Therefore, Maddy has to read 30 more pages. 6.  A herd of cows gives 4 litres of milk each day. But each cow gives one-third of total milk each day. They give 24 litres milk in six days. How many cows are there in the herd? Solution: A herd of cows gives 4 litres of milk each day. Each cow gives one-third of total milk each day = 1/3 of 4 Therefore, each cow gives 4/3 of milk each day. Total no. of cows = 4 ÷ 4/3                          = 4 × ¾                          = 3 Therefore there are 3 cows in the herd.

Questions and Answers on Word problems on Fractions:

1. Shelly walked \(\frac{1}{3}\) km. Kelly walked \(\frac{4}{15}\) km. Who walked farther? How much farther did one walk than the other?

2. A frog took three jumps. The first jump was \(\frac{2}{3}\) m long, the second was \(\frac{5}{6}\) m long and the third was \(\frac{1}{3}\) m long. How far did the frog jump in all?

3. A vessel contains 1\(\frac{1}{2}\) l of milk. John drinks \(\frac{1}{4}\) l of milk; Joe drinks \(\frac{1}{2}\) l of milk. How much of milk is left in the vessel?

●   Multiplication is Repeated Addition.

●  Multiplication of Fractional Number by a Whole Number.

●  Multiplication of a Fraction by Fraction.

●  Properties of Multiplication of Fractional Numbers.

●  Multiplicative Inverse.

●  Worksheet on Multiplication on Fraction.

●  Division of a Fraction by a Whole Number.

●  Division of a Fractional Number.

●  Division of a Whole Number by a Fraction.

●  Properties of Fractional Division.

●  Worksheet on Division of Fractions.

●  Simplification of Fractions.

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Math-M-Addicts students eagerly dive into complex math problems during class.

In the building of the Speyer Legacy School in New York City, a revolutionary math program is quietly producing some of the city's most gifted young problem solvers and logical thinkers. Founded in 2005 by two former math prodigies, Math-M-Addicts has grown into an elite academy developing the skills and mindset that traditional schooling often lacks.

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The Hechinger Report

Covering Innovation & Inequality in Education

Why schools are teaching math word problems all wrong

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CENTRAL FALLS, R.I. — When Natalia Molina began teaching her second grade students word problems earlier this school year, every lesson felt difficult. Most students were stymied by problems such as: “Sally went shopping. She spent $86 on groceries and $39 on clothing. How much more did Sally spend on groceries than on clothing?”

Both Molina, a first-year teacher, and her students had been trained to tackle word problems by zeroing in on key words like “and,” “more” and “total”  — a simplistic approach that Molina said too often led her students astray. After recognizing the word “and,” for instance, they might mistakenly assume that they needed to add two nearby numbers together to arrive at an answer.

Some weaker readers, lost in a sea of text, couldn’t recognize any words at all.

“I saw how overwhelmed they would get,” said Molina, who teaches at Segue Institute for Learning, a predominantly Hispanic charter school in this small city just north of Providence.

So, with the help of a trainer doing work in Rhode Island through a state grant, Molina and some of her colleagues revamped their approach to teaching word problems this winter — an effort that they said is already paying off in terms of increased student confidence and ability. “It has been a game changer for them,” Molina said.

Perhaps no single educational task encompasses as many different skills as the word problem. Between reading, executive functioning, problem solving, computation and vocabulary, there are a lot of ways for students to go wrong. And for that reason, students perform significantly worse overall on word problems compared to questions more narrowly focused on computation or shapes (for example: “Solve 7 + _ = 22” or “What is 64 x 3?”).

If a student excels at word problems, it’s a good sign that they’re generally excelling at school. “Word-problem solving in lower grades is one of the better indicators of overall school success in K-12,” said Lynn Fuchs, a research professor at Vanderbilt University. In a large national survey , for instance, algebra teachers rated word-problem solving as the most important among 15 skills required to excel in the subject.

Teacher takeaways

• Don’t instruct students to focus mainly on “key words” in word problems such as “and” or “more” 
• Mix question types in any lesson so that students don’t assume they just apply the same operation (addition, subtraction) again and again
• Teach students the underlying structure — or schema — of the word problem

Yet most experts and many educators agree that too many schools are doing it wrong, particularly in the elementary grades. And in a small but growing number of classrooms, teachers like Molina are working to change that. “With word problems, there are more struggling learners than non-struggling learners” because they are taught so poorly, said Nicole Bucka, who works with teachers throughout Rhode Island to provide strategies for struggling learners.

Too many teachers, particularly in the early grades, rely on key words to introduce math problems. Posters displaying the terms — sum, minus, fewer, etc. — tied to operations including addition and subtraction are a staple in elementary school classrooms across the country.

Key words can be a convenient crutch for both students and teachers, but they become virtually meaningless as the problems become harder, according to researchers. Key words can help first graders figure out whether to add or subtract more than half of the time, but the strategy rarely works for the multi-step problems students encounter starting in second and third grade. “With multi-step problems, key words don’t work 90 percent of the time,” said Sarah Powell, a professor at the University of Texas in Austin who studies word problems and whose research has highlighted the inefficacy of key words . “But the average kindergarten teacher is not thinking about that; they are teaching 5-year-olds, not 9-year-olds.”

Many teachers in the youngest grades hand out worksheets featuring the same type of word problem repeated over and over again. That’s what Molina’s colleague, Cassandra Santiago, did sometimes last year when leading a classroom on her own for the first time. “It was a mistake,” the first grade teacher said. “It’s really important to mix them up. It makes them think more critically about the parts they have to solve.”

Another flaw with word problem instruction is that the overwhelming majority of questions are divorced from the actual problem-solving a child might have to do outside the classroom in their daily life — or ever, really. “I’ve seen questions about two trains going on the same track,” said William Schmidt, a University Distinguished Professor at Michigan State University. “First, why would they be going on the same track and, second, who cares?”

Schmidt worked on an analysis of about 8,000 word problems used in 23 textbooks in 19 countries. He found that less than one percent had “real world applications” and involved “higher order math applications .”

“That is one of the reasons why children have problems with mathematics,” he said. “They don’t see the connection to the real world … We’re at this point in math right now where we are just teaching students how to manipulate numbers.”

He said a question, aimed at middle schoolers, that does have real world connections and involves more than manipulating numbers, might be: “Shopping at the new store in town includes a 43% discount on all items which are priced the same at $2. The state you live in has a 7% sales tax. You want to buy many things but only have a total of $52 to spend. Describe in words how many things you could buy.”

Schmidt added that relevancy of word problems is one area where few, if any, countries excel. “No one was a shining star leading the way,” he said. 

In her brightly decorated classroom one Tuesday afternoon, Santiago, the first grade teacher, gave each student a set of animal-shaped objects and a sheet of blue paper (the water) and green (the grass). “We’re going to work on a number story,” she told them. “I want you to use your animals to tell me the story.”

“ Once upon a time,” the story began. In this tale, three animals played in the water, and two animals played in the grass. Santiago allowed some time for the ducks, pigs and bears to frolic in the wilds of each student’s desk before she asked the children to write a number sentence that would tell them how many animals they have altogether.

Some of the students relied more on pictorial representations (three dots on one side of a line and two dots on the other) and others on the number sentence (3+2 = 5) but all of them eventually got to five. And Santiago made sure that her next question mixed up the order of operations (so students didn’t incorrectly assume that all they ever have to do is add): “Some more animals came and now there are seven. So how many more came?”

One approach to early elementary word problems that is taking off in some schools, including Segue Institute, has its origins in a special education intervention for struggling math students. Teachers avoid emphasizing key words and ask students instead to identify first the conceptual type of word problem (or schema, as many practitioners and researchers refer to it) they are dealing with: “Total problems,” for instance, involve combining two parts to find a new amount; “change problems” involve increasing or decreasing the amount of something. Total problems do not necessarily involve adding, however.

“The schemas that students learn in kindergarten will continue with them throughout their whole career,” said Powell, the word-problem researcher, who regularly works with districts across the country to help implement the approach. 

In Olathe, Kansas — a district inspired by Powell’s work — teachers had struggled for years with word problems, said Kelly Ulmer, a math support specialist whose goal is to assist in closing academic gaps that resulted from lost instruction time during the pandemic. “We’ve all tried these traditional approaches that weren’t working,” she said. “Sometimes you get pushback on new initiatives from veteran teachers and one of the things that showed us how badly this was needed is that the veteran teachers were the most excited and engaged — they have tried so many things” that haven’t worked.

In Rhode Island, many elementary schools initially used the strategy with students who required extra help, including those in special education, but expanded this use to make it part of the core instruction for all, said Bucka. In some respects, it’s similar to the recent, well publicized evolution of reading instruction in which some special education interventions for struggling readers  — most notably, a greater reliance on phonics in the early grades — have gone mainstream.

There is an extensive research bas e showing that focusing on the different conceptual types of word problems is an effective way of teaching math, although much of the research focuses specifically on students experiencing difficulties in the subject. 

Molina has found asking students to identify word problems by type to be a useful tool with nearly all of her second graders; next school year she hopes to introduce the strategy much earlier.

One recent afternoon, a lesson on word problems started with everyone standing up and chanting in unison: “Part plus part equals total” (they brought two hands together). “Total minus part equals part ” (they took one hand away) .

It’s a way to help students remember different conceptual frameworks for word problems. And it’s especially effective for the students who learn well through listening and repeating. For visual learners, the different types of word problems were mapped out on individual dry erase mats.

The real work began when Molina passed out questions, and the students— organized into the Penguin, Flower Bloom, Red Panda and Marshmallow teams — had to figure out which framework they were dealing with on their own and then work toward an answer. A few months ago, many of them would have automatically shut down when they saw the text on the page, Molina said.

For the Red Pandas, the question under scrutiny was: “The clothing store had 47 shirts. They sold 21, how many do they have now?”

“It’s a total problem,” one student said.

“No, it’s not total,” responded another.

“I think it’s about change,” said a third.

None of the students seemed worried about their lack of consensus, however. And neither was Molina. A correct answer is always nice but those come more often now that most of the students have made a crucial leap. “I notice them thinking more and more,” she said, “about what the question is actually asking.”

This story about word problems was produced by The Hechinger Report , a nonprofit, independent news organization focused on inequality and innovation in education. Sign up for the Hechinger newsletter .

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In the last example of the word problem involving the shirts would the teacher have been fine with either explanation. One thought would be that there is a total number of shirts (41) with one part being sold (21) leaving the other part on hand (26). Another way of thinking about the problem would be a change happening with the beginning number of (47) shirts being reduced or changed by the number being taken away (21), leaving the rest (26).

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Math is tripping up community college students. Some schools are trying something new

ALBANY, Ore. – It’s 7:15 on a Monday morning in May at Linn-Benton Community College in northwestern Oregon. Math professor Michael Lopez, a tape measure on his belt, paces in front of the 14 students in his “math for welders” class. “I’m your OSHA inspector,” he says. “Three-sixteenths of an inch difference, you’re in violation. You’re going to get a fine.”

He has just given them a project they might have to do on the job: figuring out the rung spacing on a steel ladder that attaches to a wall. Thousands of dollars are at stake in such builds, and they’re complicated: Some clients want the fewest possible rungs to save money; others want a specific distance between steps. To pass inspection, rungs must be evenly spaced to within one-sixteenth of an inch.

Math is a giant hurdle for most community college students pursuing welding and other career and technical degrees. About a dozen years ago, Linn-Benton’s administrators looked at their data and found that many students in career and technical education, or CTE, were getting most of the way toward a degree but were stopped by a math course, said the college’s president, Lisa Avery. That’s not unusual: Up to 60% of students entering community college are unprepared for college-level work, and the subject they most often need help with is math .

The college asked the math department to design courses tailored to those students, starting with its welding, culinary arts and criminal justice programs. The first of those, math for welders, rolled out in 2013.

More than a decade later, welding department instructors say math for welders has had a huge effect on student performance. Since 2017, 93% of students taking it have passed, and 83% have achieved all the course’s learning goals, including the ability to use arithmetic, geometry, algebra and trigonometry to solve welding problems, school data shows. Two years ago, Linn-Benton asked Lopez to design a similar course for its automotive technology program; the college began to offer that course last fall.

Math for welders changed student Zane Azmane’s view of what he could do. “I absolutely hated math in high school. It didn’t apply to anything I needed at the moment,” said Azmane, 20, who failed several semesters of math early in high school but got a B in the Linn-Benton course last year. “We actually learned equations I’m going to use, like setting ladder rungs.”

Linn-Benton’s aim is to change how students pursuing technical degrees learn math by making it directly applicable to their technical specialties.

Some researchers say these small-scale efforts to teach math in context could transform how it’s taught more broadly.

Among the strategies to help college students who struggle with math, giving them contextual curriculums seems to have "the strongest theoretical base and perhaps the strongest empirical support,” according to a 2011 paper by Dolores Perin, now professor emerita at Columbia University Teachers College. (The Hechinger Report is an independent unit of Teachers College.)

Perin’s paper echoed the results of a 2006 study of math in high school CTE involving almost 3,000 students. Students in the study who were taught math through an applied approach performed significantly better on two of three standardized tests than those taught math in a more traditional way. (The applied math students also performed better on the third test, though the results were not statistically significant.)

There haven’t been systematic studies of math in CTE at the college level, according to James Stone, director of the National Research Center for Career and Technical Education at the Southern Regional Education Board, who ran the 2006 study.

Oregon appears to be one of the few places where this approach is spreading, if slowly.

Three hours south of Linn-Benton, Doug Gardner, an instructor in the Rogue Community College math department, had long struggled with a persistent question from students: “Why do we need to know this?”

“It became my life’s work to have an answer to that question,” said Gardner, now the department chair. 

Meanwhile, at the college, about a third of students taking Algebra I or a lower-level math course failed or withdrew. For many who stayed, the lack of math knowledge hurt their job prospects, preventing them from gaining necessary skills in fields like pipefitting.

So, in 2010, Gardner applied for and won a National Science Foundation grant to create two new applied algebra courses. Instead of abstract formulas, students would learn practical ones: how to calculate the volume of a wheelbarrow of gravel and the number of wheelbarrows needed to cover an area, or how much a beam of a certain size and type would bend under a certain load.

Since then, the pass rate in the applied algebra class has averaged 73%, while the rate for the traditional course has continued to hover around 59%, according to Gardner.

One day in May, math professor Kathleen Foster was teaching applied algebra in a sun-drenched classroom on Rogue’s campus. She launched into a lesson about the Pythagorean theorem and why it’s an essential tool for building home interiors and steel structures.

James Butler-Kyniston, 30, who is pursuing a degree as a machinist, said the exercises covered in Foster’s class are directly applicable to his future career. One exercise had students calculate how large a metal sheet you would need to manufacture a certain number of parts at a time. “Algebraic formulas apply to a lot of things, but since you don’t have any examples to tie them to, you end up thinking they’re useless,” he said.

In 2021, Oregon state legislators passed a law requiring all four-year colleges to accept an applied math community college course called Math in Society as satisfying the math requirement for a four-year degree. In that course, instead of studying theoretical algebra, students learn how to use probability and statistics to interpret the results in scientific papers and how political rules like apportionment and gerrymandering affect elections, said Kathy Smith, a math professor at Central Oregon Community College.

“If I had my way, this is how algebra would be taught to every student: the applied version,” Gardner said. “And then if a student says ‘This is great, but I want to go further,’ then you sign up for the theoretical version.”

But at individual schools, lack of money and time constrain the spread of applied math. Stone’s team works with high schools around the country to design contextual math courses for career and technical students. They tried to work with a few community colleges, but their CTE faculty, many of whom were part-timers on contract, didn’t have time to partner with their math departments to develop a new curriculum, a yearlong process, Stone said.

Linn-Benton was able to invest the time and money because its math department was big enough to take on the task, said Avery. Both Linn-Benton and Rogue may be outliers because they have math faculty with technical backgrounds: Lopez worked as a carpenter and sheriff’s deputy and served three tours as a machine gunner in Iraq, and Gardner was a construction contractor.

Back in Lopez’s class, students are done calculating where their ladder rungs should go and now must mark them on the wall.

As teams finish up, Lopez inspects their work. “That’s one-thirty-second shy. But I wouldn’t worry too much about it,” he tells one group. “OSHA’s not going to knock you down for that.”

Three teams pass, two fail – but this is the place to make mistakes, not out on the job, Lopez tells them.

“This stuff is hard,” said Keith Perkins, 40, who’s going for a welding degree and wants to get into the local pipe fitters union. “I hated math in school. Still hate it. But we use it every day.”

This story was produced by The Hechinger Report , a nonprofit, independent news organization focused on inequality and innovation in education.

Fraction Word Problems (Difficult)

Here are some examples of more difficult fraction word problems. We will illustrate how block models (tape diagrams) can be used to help you to visualize the fraction word problems in terms of the information given and the data that needs to be found.

Related Pages Fraction Word Problems Singapore Math Lessons Fraction Problems Using Algebra Algebra Word Problems

Block modeling (also known as tape diagrams or bar models) are widely used in Singapore Math and the Common Core to help students visualize and understand math word problems.

Example: 2/9 of the people on a restaurant are adults. If there are 95 more children than adults, how many children are there in the restaurant?

Solution: Draw a diagram with 9 equal parts: 2 parts to represent the adults and 7 parts to represent the children.

5 units = 95 1 unit = 95 ÷ 5 = 19 7 units = 7 × 19 = 133

Answer: There are 133 children in the restaurant.

Example: Gary and Henry brought an equal amount of money for shopping. Gary spent $95 and Henry spent $350. After that Henry had 4/7 of what Gary had left. How much money did Gary have left after shopping?

350 – 95 = 255 3 units = 255 1 unit = 255 ÷ 3 = 85 7 units = 85 × 7 = 595

Answer: Gary has $595 after shopping.

Example: 1/9 of the shirts sold at Peter’s shop are striped. 5/8 of the remainder are printed. The rest of the shirts are plain colored shirts. If Peter’s shop has 81 plain colored shirts, how many more printed shirts than plain colored shirts does the shop have?

Solution: Draw a diagram with 9 parts. One part represents striped shirts. Out of the remaining 8 parts: 5 parts represent the printed shirts and 3 parts represent plain colored shirts.

3 units = 81 1 unit = 81 ÷ 3 = 27 Printed shirts have 2 parts more than plain shirts. 2 units = 27 × 2 = 54

Answer: Peter’s shop has 54 more printed colored shirts than plain shirts.

Solve a problem involving fractions of fractions and fractions of remaining parts

Example: 1/4 of my trail mix recipe is raisins and the rest is nuts. 3/5 of the nuts are peanuts and the rest are almonds. What fraction of my trail mix is almonds?

How to solve fraction word problem that involves addition, subtraction and multiplication using a tape diagram or block model

Example: Jenny’s mom says she has an hour before it’s bedtime. Jenny spends 3/5 of the hour texting a friend and 3/8 of the remaining time brushing her teeth and putting on her pajamas. She spends the rest of the time reading her book. How long did Jenny read?

How to solve a four step fraction word problem using tape diagrams?

Example: In an auditorium, 1/6 of the students are fifth graders, 1/3 are fourth graders, and 1/4 of the remaining students are second graders. If there are 96 students in the auditorium, how many second graders are there?

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The Supreme Court says cities can punish people for sleeping in public places

Jennifer Ludden

U.S. Supreme Court says cities can punish people for sleeping in public places

A homeless person walks near an elementary school in Grants Pass, Ore., on March 23. The rural city became the unlikely face of the nation's homelessness crisis when it asked the U.S. Supreme Court to uphold its anti-camping laws. Jenny Kane/AP hide caption

In its biggest decision on homelessness in decades, the U.S. Supreme Court today ruled that cities can ban people from sleeping and camping in public places. The justices, in a 6-3 decision along ideological lines, overturned lower court rulings that deemed it cruel and unusual under the Eighth Amendment to punish people for sleeping outside if they had nowhere else to go.

Writing for the majority, Justice Gorsuch said, “Homelessness is complex. Its causes are many.” But he said federal judges do not have any “special competence” to decide how cities should deal with this.

“The Constitution’s Eighth Amendment serves many important functions, but it does not authorize federal judges to wrest those rights and responsibilities from the American people and in their place dictate this Nation’s homelessness policy,” he wrote.

In a dissent, Justice Sotomayor said the decision focused only on the needs of cities but not the most vulnerable. She said sleep is a biological necessity, but this decision leaves a homeless person with “an impossible choice — either stay awake or be arrested.”

The court's decision is a win not only for the small Oregon city of Grants Pass, which brought the case, but also for dozens of Western localities that had urged the high court to grant them more enforcement powers as they grapple with record high rates of homelessness. They said the lower court rulings had tied their hands in trying to keep public spaces open and safe for everyone.

Supreme Court appears to side with an Oregon city's crackdown on homelessness

But advocates for the unhoused say the decision won’t solve the bigger problem, and could make life much harder for the quarter of a million people living on streets, in parks and in their cars. “Where do people experiencing homelessness go if every community decides to punish them for their homelessness?” says Diane Yentel, president of the National Low Income Housing Coalition.

Today’s ruling only changes current law in the 9th Circuit Court of Appeals, which includes California and eight other Western states where the bulk of America’s unhoused population lives. But it will also determine whether similar policies elsewhere are permissible; and it will almost certainly influence homelessness policy in cities around the country.

Cities complained they were hamstrung in managing a public safety crisis

Grants Pass and other cities argued that lower court rulings fueled the spread of homeless encampments, endangering public health and safety. Those decisions did allow cities to restrict when and where people could sleep and even to shut down encampments – but they said cities first had to offer people adequate shelter.

That’s a challenge in many places that don’t have nearly enough shelter beds. In briefs filed by local officials, cities and town also expressed frustration that many unhoused people reject shelter when it is available; they may not want to go if a facility bans pets, for example, or prohibits drugs and alcohol.

Critics also said lower court rulings were ambiguous, making them unworkable in practice. Localities have faced dozens of lawsuits over the details of what’s allowed. And they argued that homelessness is a complex problem that requires balancing competing interests, something local officials are better equipped to do than the courts.

"We are trying to show there's respect for the public areas that we all need to have," Seattle City Attorney Ann Davison told NPR earlier this year. She wrote a legal brief on behalf of more than a dozen other cities. "We care for people, and we're engaging and being involved in the long-term solution for them."

The decision will not solve the larger problem of rising homelessness

Attorneys for homeless people in Grants Pass argued that the city’s regulations were so sweeping, they effectively made it illegal for someone without a home to exist. To discourage sleeping in public spaces, the city banned the use of stoves and sleeping bags, pillows or other bedding. But Grants Pass has no public shelter, only a Christian mission that imposes various restrictions and requires people to attend religious service.

"It's sort of the bare minimum in what a just society should expect, is that you're not going to punish someone for something they have no ability to control," said Ed Johnson of the Oregon Law Center, which represents those who sued the city.

He also said saddling people with fines and a criminal record makes it even harder for them to eventually get into housing.

Johnson and other advocates say today’s decision won’t change the core problem behind rising homelessness: a severe housing shortage, and rents that have become unaffordable for a record half of all tenants. The only real solution, they say, is to create lots more housing people can afford – and that will take years.

• homelessness
• Supreme Court

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